Chapter 11 trigonometric identities
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Chapter 11: Trigonometric Identities. 11.1 Trigonometric Identities 11.2 Addition and Subtraction Formulas 11.3 Double-Angle, Half-Angle, and Product-Sum Formulas 11.4 Inverse Trigonometric Functions 11.5 Trigonometric Equations. 11.2 Sum and Difference Identities.

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Chapter 11: Trigonometric Identities

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Chapter 11 trigonometric identities

Chapter 11: Trigonometric Identities

11.1Trigonometric Identities

11.2Addition and Subtraction Formulas

11.3Double-Angle, Half-Angle, and Product-Sum Formulas

11.4Inverse Trigonometric Functions

11.5Trigonometric Equations


11 2 sum and difference identities

11.2Sum and Difference Identities

Derive the identity for cos(A – B).

Let angles A and B be angles in standard position on a unit circle with B < A.

Let Sand Q be the points on the terminal sides of angels A and B, respectively.

Q has coordinates (cosB, sin B).

S has coordinates (cosA, sin A).

R has coordinates (cos (A – B), sin (A – B)).

Angle SOQ equals A – B.

Since SOQ = POR, chords PR and SQ are equal.


11 2 sum and difference identities1

11.2Sum and Difference Identities

  • By the distance formula, chords PR = SQ,

    Simplifying this equation and using the identity

    sin² x + cos² x =1, we rewrote the equation as

    cos(A– B) = cosAcosB + sin A sin B.


11 2 sum and difference identities2

11.2 Sum and Difference Identities

  • To find cos(A + B), rewrite A + B as A– (–B) and use the identity for cos (A–B).

Cosine of a Sum Or Difference

cos(A – B) = cos A cos B + sin A sin B

cos(A + B) = cos A cos B– sin A sin B


11 2 finding exact cosine values

11.2Finding Exact Cosine Values

ExampleFind the exact value of the following.

  • cos 15°


11 2 sine of a sum or difference

11.2Sine of a Sum or Difference

Using the cofunction relationship and letting = A + B, we can obtain the:

Sine of a Sum or Difference

sin(A + B) = sin A cosB + cosA sin B

sin(A–B) = sin A cosB –cosA sin B


11 2 tangent of a sum or difference

11.2Tangent of a Sum or Difference

  • Using the identities for sin(A+ B), cos(A+ B), and tan(–B) = –tan B, we can derive the identities for the tangent of a sum or difference.

Tangent of a Sum or Difference


11 2 example using sine and tangent sum or difference formulas

11.2Example Using Sine and Tangent Sum or Difference Formulas

ExampleFind the exact value of the following.

  • sin 75°

  • tan

  • sin 40° cos 160° – cos 40° sin 160°

    Solution

    (a)


11 2 example using sine and tangent sum or difference formulas1

11.2Example Using Sine and Tangent Sum or Difference Formulas

(b)

(c) sin 40° cos 160° – cos 40° sin 160°

= sin(40° – 160°)

= sin(–120°)


11 2 finding function values and the quadrant of a b

11.2Finding Function Values and the Quadrant of A + B

ExampleSuppose that A and B are angles in standard

position, with

Find each of the following.

  • sin(A + B) (b) tan (A + B) (c) the quadrant of A + B

    Solution

    (a)

Since cosA < 0 in Quadrant II.


11 2 finding function values and the quadrant of a b1

11.2Finding Function Values and the Quadrant of A + B

(b) Use the values of sine and cosine from part (a) to

get

(c) From the results of parts (a) and (b), we find that

sin(A + B) is positive and tan(A + B) is also positive. Therefore, A + B must be in quadrant I.


11 2 applying the cosine difference identity to voltage

11.2Applying the Cosine Difference Identity to Voltage

ExampleCommon household electric current is called

alternating current because the current alternates direction

within the wire. The voltage V in a typical 115-volt outlet can be

expressed by the equation V = 163 sin t, where  is the

angular velocity (in radians per second) of the rotating generator

at the electrical plant and t is time measured in seconds.

  • It is essential for electric generators to rotate at 60 cycles per second so household appliances and computers will function properly. Determine  for these electric generators.

  • Graph V on the interval 0  t  .05.

  • For what value of  will the graph of V = 163cos(t–) be the same as the graph of V = 163 sin t?


11 2 applying the cosine difference identity to voltage1

11.2Applying the Cosine Difference Identity to Voltage

Solution

  • Since each cycle is 2 radians, at 60 cycles per second,

     = 60(2) = 120 radians per second.

    (b)V = 163 sin t = 163 sin 120t.

    Because amplitude is 163,

    choose –200  V  200 for the

    range.

    (c)


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